Bartlett et al. [1]} investigate the VC dimension of feedforward neural networks with piecewise polynomial activation functions. A piecewise polynimial activation function with \(p\)  pieces has the form \(\sigma (z) = \phi _i(z)\) , where \(z\in [t_{i-1}, t_i)\) , \(i \in {1,\ldots ,p+1}\) , and \(t_{i-1}< t_{i}\) . Each \(\phi _i\)  is a polynomial function of degree no more than \(r\) . Let \(W\)  be the number of parameters in the network and \(L\)  be the depth of the network. Bartlett et al. [1]} prove that the upper bound for the VC dimension of such networks is \(O(WL^2+WL\log WL)\)  and the lower bound for the VC dimension is \(\Omega (WL)\) . Later, Bartlett et al. [3]} improve this lower bound to \(\Omega (WL\log (W/L))\) .
